I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of fiber bundles and connections that are rigorous but at the same time gives what we need to apply in Physics?
I think I good book for that may be C. J. Isham's Modern Differential Geometry for Physicists. I haven't gotten to the chapter of fiber bundles, but what I've read seems to be quite rigorous. And as it is written for physicists, I think it could please your needs.
Nakahara's Geometry, Topology and Physics has two chapters covering Fiber Bundles up to Connections on Fiber Bundles with a few applications in Gauge theories. If it is your first time learning Fiber Bundles i would recommend this books, it's rigorous and has a lot of physics-motivated examples.